Assume that the market is currently supplied by ten identical sellers each with variable cost VC(𝑞) = 3𝑞 + 1/4𝑞^2, and the market demand is 𝐷 = 10(7 − 𝑝). Assume that the market demand is sufficiently high that no seller shuts down.(a) [5 marks ]Find each seller’s marginal cost curve.(b) [10 marks ]Assume that there are sufficiently many sellers so that the market can be treated as perfectly competitive. Find the competitive equilibrium.
Question
Assume that the market is currently supplied by ten identical sellers each with variable cost VC(𝑞) = 3𝑞 + 1/4𝑞^2, and the market demand is 𝐷 = 10(7 − 𝑝). Assume that the market demand is sufficiently high that no seller shuts down.(a) [5 marks ]Find each seller’s marginal cost curve.(b) [10 marks ]Assume that there are sufficiently many sellers so that the market can be treated as perfectly competitive. Find the competitive equilibrium.
Solution 1
(a) To find the marginal cost curve, we need to take the derivative of the variable cost function with respect to quantity (q).
The variable cost function is VC(q) = 3q + 1/4q^2.
The derivative of this function is MC(q) = d(VC)/dq = 3 + 1/2q.
So, the marginal cost curve for each seller is MC(q) = 3 + 1/2q.
(b) In a perfectly competitive market, the price is determined by the intersection of the market demand and supply curves.
The market demand curve is D = 10(7 - p). We can rewrite this as p = 7 - D/10.
The market supply curve is determined by the marginal cost curve of the sellers. Since there are 10 identical sellers, the market supply is 10 times the quantity supplied by an individual seller.
In a perfectly competitive market, a seller sets quantity such that price equals marginal cost. So, we set p = MC(q) to find the quantity supplied by an individual seller:
7 - D/10 = 3 + 1/2q => D/10 = 4 - 1/2q => D = 40 - 5q
Since D is the total quantity demanded and 10q is the total quantity supplied (because there are 10 sellers), we set D = 10q to find the equilibrium quantity:
40 - 5q = 10q => 15q = 40 => q = 40/15 = 8/3
Substituting q = 8/3 into the demand function gives the equilibrium price:
p = 7 - D/10 = 7 - (40 - 5*8/3)/10 = 7 - (40 - 40/3)/10 = 7 - 8/3 = 13/3
So, the competitive equilibrium is a price of 13/3 and a quantity of 8/3 for each seller.
Solution 2
(a) To find the marginal cost curve, we need to take the derivative of the variable cost function with respect to quantity (q).
The variable cost function is VC(q) = 3q + 1/4q^2.
The derivative of this function is MC(q) = d(VC)/dq = 3 + 1/2q.
So, the marginal cost curve for each seller is MC(q) = 3 + 1/2q.
(b) In a perfectly competitive market, the price is determined by the intersection of the market demand and supply curves.
The market demand curve is D = 10(7 - p). We can rewrite this as p = 7 - D/10.
The market supply curve is determined by the marginal cost curve of the sellers. Since there are 10 identical sellers, the market supply is 10 times the quantity supplied by an individual seller.
In a perfectly competitive market, a seller sets quantity such that price equals marginal cost. So, we set p = MC(q) to find the quantity supplied by an individual seller:
7 - D/10 = 3 + 1/2q => D/10 = 4 - 1/2q => D = 40 - 5q
Since D is the total quantity demanded and 10q is the total quantity supplied (because there are 10 sellers), we set D = 10q to find the equilibrium quantity:
40 - 5q = 10q => 15q = 40 => q = 40/15 = 8/3
Substituting q = 8/3 into the demand function gives the equilibrium price:
p = 7 - D/10 = 7 - (40 - 5*8/3)/10 = 7 - (40 - 40/3)/10 = 7 - 8/3 = 13/3
So, the competitive equilibrium is a price of 13/3 and a quantity of 8/3 for each seller.
Similar Questions
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